If my history is correct, Rav Natan lived before the Bavli was compiled. It seems that in a sense, Avot D'Rav Natan is a type of "tosefta". There are numerous midrashim and quotations in the Gemarrah that are duplicated in Avot D'Rav Natan. I would think the Gemarrah wouls cross reference the source in some way, but I haven't noticed it. At least, perhaps the Gemarrah would use a phrase like "Ta Shma" or "Amar Rav Natan" or something like that. Am I missing something in the lack of the Gemarrah's cross reference?
Avot deRabbi Natan was actually compiled in the Geonic era, as a sort of gemara on Pirkei Avot:
Avot de-Rabbi Nathan (Hebrew: אבות דרבי נתן), usually printed together with the minor tractates of the Talmud, is a Jewish aggadic work probably compiled in the geonic era (c.700–900 CE). Although Avot de-Rabbi Nathan is the first and longest of the "minor tractates", it probably does not belong in that collection chronologically, having more the character of a late midrash. In the form now extant it contains a mixture of Mishnah and Midrash, and may be technically designated as a homiletical exposition of the Mishnaic tractate Pirkei Avoth, having for its foundation an older recension (version) of that tractate.
If so, it is quite probable that much of the material there which also appears in the gemara was actually drawn from the gemara. If so, we would not expect the gemara to reference Avot deRabbi Natan in any of those instances.
The end of Maseches Horiyos records the story that Rebbe Meir and Rebbi Nosson attempted to oust Rabban Shimon Ben Gamliel from his position of Nasi. After they failed, Rabban Shimon "fined" Rebbi Nosson that his name should not be mentioned any more in the beis medrash. Therefore, his opinion is cited as "יש אומרים" - "some say."
אמר ליה ר"מ לרבי נתן אנא חכם ואת אב"ד נתקין מילתא כי לדידן מאי נעביד ליה נימא ליה גלי עוקצים דלית ליה וכיון דלא גמר נימא ליה (תהילים קו) מי ימלל גבורות ה' ישמיע כל תהלתו למי נאה למלל גבורות ה' מי שיכול להשמיע כל תהלותיו נעבריה והוי אנא אב"ד ואת נשיא
אמר להן רבן [שמעון בן] גמליאל ניעיילינהו מיהו ניקנסינהו דלא נימרו שמעתא משמייהו אסיקו לרבי מאיר אחרים ולר' נתן יש אומרים
For the same reason, it is possible that his Aggadic teachings were not referenced by name when quoted in the Talmud.